Find the equation of the circle which touches the y-axis and whose centre is (1, 3).
Understand the Problem
The question is asking to find the equation of a circle that touches the y-axis and has a center at the point (1, 3). We will need to use the standard form of the circle's equation and consider the condition of tangency to the y-axis.
Answer
The equation of the circle is $$(x - 1)^2 + (y - 3)^2 = 1.$$
Answer for screen readers
The equation of the circle is $$ (x - 1)^2 + (y - 3)^2 = 1 $$
Steps to Solve
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Identify the center and radius of the circle
The center of the circle is given as ( (1, 3) ). Since the circle touches the y-axis, the radius is the horizontal distance from the center to the y-axis. This distance is equal to the x-coordinate of the center, which is 1. Thus, the radius ( r ) is 1. -
Use the standard form of a circle’s equation
The standard equation of a circle with center ( (h, k) ) and radius ( r ) is given by: $$ (x - h)^2 + (y - k)^2 = r^2 $$
Here, ( h = 1 ), ( k = 3 ), and ( r = 1 ). -
Substitute the values into the equation
Plugging in the values gives: $$ (x - 1)^2 + (y - 3)^2 = 1^2 $$
This simplifies to: $$ (x - 1)^2 + (y - 3)^2 = 1 $$ -
Final equation
The final equation of the circle is presented as: $$ (x - 1)^2 + (y - 3)^2 = 1 $$
The equation of the circle is $$ (x - 1)^2 + (y - 3)^2 = 1 $$
More Information
This equation represents a circle with a center at ( (1, 3) ) and a radius of 1. Since the circle touches the y-axis, it confirms that the distance from the center to the y-axis equals the radius.
Tips
- Confusing the radius: Sometimes, students mistakenly calculate the radius as the y-coordinate instead of the x-coordinate when the condition is of tangency to the y-axis.
- Forgetting to square the radius: Ensure that the radius is squared in the equation; forgetting this can lead to wrong answers.
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